FIND THE ERROR Your classmate says that any two congruent acute triangles can be arranged to make a rectangle. Describe the error. What support can you provide to help your classmate correct the error?

Your classmate's statement that "any two congruent acute triangles can be arranged to make a rectangle" contains a fundamental geometric error related to the angles and properties required to form a rectangle. Here's a detailed explanation of the error and how to address it: ### **Identifying the Error** 1. **Understanding Triangle Types:** - **Acute Triangle:** All three interior angles are less than 90 degrees. - **Right Triangle:** One interior angle is exactly 90 degrees. - **Obtuse Triangle:** One interior angle is greater than 90 degrees. 2. **Forming a Rectangle:** - A rectangle is a quadrilateral with four right angles (each angle is 90 degrees). - To form a rectangle by arranging two triangles, the angles at the joint where the triangles meet must sum to 90 degrees. 3. **Why Acute Triangles Don’t Work:** - In two congruent acute triangles, all angles are less than 90 degrees. - When you attempt to pair two acute angles from each triangle to form a corner of a rectangle, their sum will be less than 180 degrees, not 90 degrees. - This means the resulting figure will not have the necessary right angles to qualify as a rectangle. 4. **Correct Scenario with Right Triangles:** - If you use two congruent **right** triangles, you can arrange them such that their right angles form the corners of a rectangle. - This works because the right angles complement each other to meet the 90-degree requirement at each corner of the rectangle. ### **Supporting Your Classmate to Correct the Error** 1. **Provide Visual Aids:** - **Draw Two Acute Triangles:** Show that when you place them together, the angles where they meet do not add up to 90 degrees. - **Draw Two Right Triangles:** Demonstrate how their right angles align perfectly to form a rectangle. 2. **Mathematical Explanation:** - **Angle Sum:** Explain that for a rectangle, each corner requires angles that sum to exactly 90 degrees. - **Acute Angle Limitation:** Since each angle in an acute triangle is less than 90 degrees, combining two such angles cannot achieve the necessary 90 degrees. 3. **Counterexample:** - Present a specific example with measurements. For instance, take two congruent acute triangles with angles 30°, 60°, and 90° (though technically one angle is 90°, contradicting the "acute" definition) to show that without a right angle, forming a rectangle isn't possible. - Alternatively, use truly acute triangles (e.g., 50°, 60°, 70°) and demonstrate that their combined angles at any joint do not meet the rectangle's requirements. 4. **Clarify Misconceptions:** - Ensure your classmate understands the difference between acute and right triangles. - Emphasize that only specific types of triangles (like right triangles) can be used to form rectangles when paired correctly. ### **Conclusion** The primary error in the statement lies in assuming that any two congruent acute triangles can meet the angle requirements necessary to form a rectangle. By understanding the properties of acute versus right triangles and how angles sum up in geometric figures, it's clear that only certain triangle types (specifically right triangles) can be arranged to form rectangles. Providing visual examples, mathematical explanations, and counterexamples will help your classmate grasp why the original statement is incorrect and how to correct their understanding.